Properties

Label 1900.2.i.d
Level $1900$
Weight $2$
Character orbit 1900.i
Analytic conductor $15.172$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 9 x^{6} + 2 x^{5} + 65 x^{4} - 20 x^{3} + 25 x^{2} + 6 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{7} q^{7} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{7} q^{7} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{9} + ( 1 + 2 \beta_{3} - \beta_{7} ) q^{11} + ( -2 + \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{13} + ( -\beta_{1} - \beta_{6} ) q^{17} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{19} + ( -2 \beta_{4} + \beta_{6} ) q^{21} + ( -\beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{23} + ( -2 + \beta_{2} + 2 \beta_{3} + \beta_{7} ) q^{27} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{29} + ( -3 + \beta_{2} - 2 \beta_{3} + \beta_{7} ) q^{31} + ( -\beta_{1} + 6 \beta_{4} - \beta_{6} ) q^{33} + ( 6 - 2 \beta_{3} + \beta_{7} ) q^{37} + ( -6 + 2 \beta_{2} + 3 \beta_{3} ) q^{39} + ( 2 \beta_{4} + \beta_{5} ) q^{41} + ( \beta_{1} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{43} + ( -4 + 2 \beta_{2} - 4 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{47} + ( 3 + \beta_{2} + 2 \beta_{3} - \beta_{7} ) q^{49} + ( 2 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{51} + ( -2 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{53} + ( -4 - 3 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{57} + ( -\beta_{1} - 3 \beta_{4} + 2 \beta_{5} ) q^{59} + ( 3 + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{61} + ( 2 + 5 \beta_{1} + 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{63} + ( 2 \beta_{2} + 2 \beta_{6} ) q^{67} + ( -\beta_{2} - 3 \beta_{3} - \beta_{7} ) q^{69} + ( 2 \beta_{1} - 3 \beta_{4} - \beta_{6} ) q^{71} + ( 4 \beta_{1} + \beta_{5} + \beta_{6} ) q^{73} + ( 6 + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{7} ) q^{77} + ( \beta_{1} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{79} + ( -4 \beta_{1} + 5 \beta_{4} + \beta_{5} ) q^{81} + ( -2 - \beta_{2} - 3 \beta_{3} ) q^{83} + ( \beta_{2} - 2 \beta_{3} ) q^{87} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{89} + ( -12 - 2 \beta_{1} - 2 \beta_{3} - 12 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} ) q^{91} + ( -4 \beta_{1} - 8 \beta_{4} + \beta_{5} + \beta_{6} ) q^{93} + ( \beta_{1} + 2 \beta_{6} ) q^{97} + ( 5 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} + 5 \beta_{4} + 4 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + q^{3} - 5q^{9} + O(q^{10}) \) \( 8q + q^{3} - 5q^{9} + 4q^{11} - 9q^{13} - q^{17} + 3q^{19} + 8q^{21} - 20q^{27} + 5q^{29} - 20q^{31} - 25q^{33} + 52q^{37} - 54q^{39} - 8q^{41} - 7q^{43} - 16q^{47} + 20q^{49} + 12q^{51} - 5q^{53} - 27q^{57} + 11q^{59} + 12q^{61} + 3q^{63} + 6q^{69} + 14q^{71} + 4q^{73} + 44q^{77} + 13q^{79} - 24q^{81} - 10q^{83} + 4q^{87} + 5q^{89} - 46q^{91} + 28q^{93} + q^{97} + 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 9 x^{6} + 2 x^{5} + 65 x^{4} - 20 x^{3} + 25 x^{2} + 6 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -304 \nu^{7} - 31 \nu^{6} - 2546 \nu^{5} - 1710 \nu^{4} - 23348 \nu^{3} - 1178 \nu^{2} - 380 \nu + 48866 \)\()/13629\)
\(\beta_{3}\)\(=\)\((\)\( 536 \nu^{7} - 304 \nu^{6} + 4489 \nu^{5} + 3015 \nu^{4} + 36145 \nu^{3} + 2077 \nu^{2} + 670 \nu + 3506 \)\()/13629\)
\(\beta_{4}\)\(=\)\((\)\( 1753 \nu^{7} - 2825 \nu^{6} + 16385 \nu^{5} - 5472 \nu^{4} + 107915 \nu^{3} - 107350 \nu^{2} + 39671 \nu - 18080 \)\()/27258\)
\(\beta_{5}\)\(=\)\((\)\( 1418 \nu^{7} - 2635 \nu^{6} + 15283 \nu^{5} - 9060 \nu^{4} + 100657 \nu^{3} - 100130 \nu^{2} + 131248 \nu - 16864 \)\()/13629\)
\(\beta_{6}\)\(=\)\((\)\( 3274 \nu^{7} - 5315 \nu^{6} + 30827 \nu^{5} - 12249 \nu^{4} + 203033 \nu^{3} - 201970 \nu^{2} + 65423 \nu - 34016 \)\()/13629\)
\(\beta_{7}\)\(=\)\((\)\( -1328 \nu^{7} + 821 \nu^{6} - 11122 \nu^{5} - 7470 \nu^{4} - 84061 \nu^{3} - 5146 \nu^{2} - 1660 \nu - 16552 \)\()/4543\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - 4 \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 8 \beta_{3} + \beta_{2} - 2\)
\(\nu^{4}\)\(=\)\(-9 \beta_{6} + \beta_{5} + 32 \beta_{4} - 13 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-9 \beta_{7} - 14 \beta_{6} + 9 \beta_{5} + 36 \beta_{4} - 72 \beta_{3} - 14 \beta_{2} - 72 \beta_{1} + 36\)
\(\nu^{6}\)\(=\)\(-14 \beta_{7} - 150 \beta_{3} - 81 \beta_{2} + 278\)
\(\nu^{7}\)\(=\)\(164 \beta_{6} - 81 \beta_{5} - 466 \beta_{4} + 671 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(1\) \(-1 - \beta_{4}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
−1.26041 + 2.18309i
−0.176725 + 0.306096i
0.354609 0.614201i
1.58253 2.74101i
−1.26041 2.18309i
−0.176725 0.306096i
0.354609 + 0.614201i
1.58253 + 2.74101i
0 −1.26041 + 2.18309i 0 0 0 −2.72743 0 −1.67727 2.90511i 0
201.2 0 −0.176725 + 0.306096i 0 0 0 4.30507 0 1.43754 + 2.48989i 0
201.3 0 0.354609 0.614201i 0 0 0 −3.11079 0 1.24850 + 2.16247i 0
201.4 0 1.58253 2.74101i 0 0 0 1.53315 0 −3.50877 6.07738i 0
501.1 0 −1.26041 2.18309i 0 0 0 −2.72743 0 −1.67727 + 2.90511i 0
501.2 0 −0.176725 0.306096i 0 0 0 4.30507 0 1.43754 2.48989i 0
501.3 0 0.354609 + 0.614201i 0 0 0 −3.11079 0 1.24850 2.16247i 0
501.4 0 1.58253 + 2.74101i 0 0 0 1.53315 0 −3.50877 + 6.07738i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.i.d 8
5.b even 2 1 380.2.i.c 8
5.c odd 4 2 1900.2.s.d 16
15.d odd 2 1 3420.2.t.w 8
19.c even 3 1 inner 1900.2.i.d 8
20.d odd 2 1 1520.2.q.m 8
95.h odd 6 1 7220.2.a.p 4
95.i even 6 1 380.2.i.c 8
95.i even 6 1 7220.2.a.r 4
95.m odd 12 2 1900.2.s.d 16
285.n odd 6 1 3420.2.t.w 8
380.p odd 6 1 1520.2.q.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.c 8 5.b even 2 1
380.2.i.c 8 95.i even 6 1
1520.2.q.m 8 20.d odd 2 1
1520.2.q.m 8 380.p odd 6 1
1900.2.i.d 8 1.a even 1 1 trivial
1900.2.i.d 8 19.c even 3 1 inner
1900.2.s.d 16 5.c odd 4 2
1900.2.s.d 16 95.m odd 12 2
3420.2.t.w 8 15.d odd 2 1
3420.2.t.w 8 285.n odd 6 1
7220.2.a.p 4 95.h odd 6 1
7220.2.a.r 4 95.i even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 4 + 6 T + 25 T^{2} - 20 T^{3} + 65 T^{4} + 2 T^{5} + 9 T^{6} - T^{7} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 56 - 11 T - 19 T^{2} + T^{4} )^{2} \)
$11$ \( ( 267 + 21 T - 35 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$13$ \( 2704 - 6708 T + 16901 T^{2} - 291 T^{3} + 1134 T^{4} + 213 T^{5} + 86 T^{6} + 9 T^{7} + T^{8} \)
$17$ \( 36864 + 1728 T + 6609 T^{2} - 690 T^{3} + 955 T^{4} - 52 T^{5} + 35 T^{6} + T^{7} + T^{8} \)
$19$ \( 130321 - 20577 T - 15523 T^{2} + 399 T^{3} + 1227 T^{4} + 21 T^{5} - 43 T^{6} - 3 T^{7} + T^{8} \)
$23$ \( 2916 - 5346 T + 7695 T^{2} - 3861 T^{3} + 1575 T^{4} - 198 T^{5} + 39 T^{6} + T^{8} \)
$29$ \( 74529 - 86814 T + 85017 T^{2} - 21492 T^{3} + 5344 T^{4} - 341 T^{5} + 84 T^{6} - 5 T^{7} + T^{8} \)
$31$ \( ( -277 - 303 T - 29 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$37$ \( ( 414 - 609 T + 217 T^{2} - 26 T^{3} + T^{4} )^{2} \)
$41$ \( 324 - 594 T + 999 T^{2} - 453 T^{3} + 271 T^{4} + 106 T^{5} + 59 T^{6} + 8 T^{7} + T^{8} \)
$43$ \( 31684 + 31506 T + 25633 T^{2} + 8156 T^{3} + 2441 T^{4} + 130 T^{5} + 81 T^{6} + 7 T^{7} + T^{8} \)
$47$ \( 1382976 + 1160712 T + 944769 T^{2} + 62307 T^{3} + 17593 T^{4} + 1574 T^{5} + 281 T^{6} + 16 T^{7} + T^{8} \)
$53$ \( 1483524 - 734454 T + 507333 T^{2} + 58974 T^{3} + 15721 T^{4} + 616 T^{5} + 143 T^{6} + 5 T^{7} + T^{8} \)
$59$ \( 2518569 + 733194 T + 300729 T^{2} + 9504 T^{3} + 6520 T^{4} - 319 T^{5} + 176 T^{6} - 11 T^{7} + T^{8} \)
$61$ \( 841 + 12644 T + 190734 T^{2} - 8896 T^{3} + 5687 T^{4} - 608 T^{5} + 166 T^{6} - 12 T^{7} + T^{8} \)
$67$ \( 10863616 + 184576 T + 411840 T^{2} - 6944 T^{3} + 12080 T^{4} - 112 T^{5} + 124 T^{6} + T^{8} \)
$71$ \( 5049009 - 1503243 T + 445314 T^{2} - 63585 T^{3} + 11614 T^{4} - 1324 T^{5} + 197 T^{6} - 14 T^{7} + T^{8} \)
$73$ \( 311364 - 41850 T + 82071 T^{2} + 14739 T^{3} + 17911 T^{4} + 698 T^{5} + 153 T^{6} - 4 T^{7} + T^{8} \)
$79$ \( 9096256 - 5464992 T + 2897296 T^{2} - 310352 T^{3} + 42956 T^{4} - 1960 T^{5} + 297 T^{6} - 13 T^{7} + T^{8} \)
$83$ \( ( -1302 - 669 T - 82 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$89$ \( 1382976 + 98784 T + 150528 T^{2} + 1512 T^{3} + 14128 T^{4} + 442 T^{5} + 147 T^{6} - 5 T^{7} + T^{8} \)
$97$ \( 13351716 + 186354 T + 448389 T^{2} + 1086 T^{3} + 11281 T^{4} + 20 T^{5} + 123 T^{6} - T^{7} + T^{8} \)
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